On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

In this paper we analyze the porous medium equation \begin{equation}\label{ProblemAbstract} \tag{$\mathrm{◊}$} %\begin{cases} u_t=\Delta u^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad $\text{in}$\quad \Omega \times I,%\ %u_\nu-g(u)=0 & $\text{on}$; \partial \Omega, t>0,\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\ %\end{cases} \end{equation} where $\mathrm{\Omega }$ is a bounded and smooth domain of ${\mathbb{R}}^N$, with $N\ge 1$, and $I=[0,t^{\ast })$ is the maximal interval of existence for $u$. The constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $u$. Under some hypothesis on the data, including intrinsic relations between $m,p$ and $q$, and assuming that for some positive and sufficiently regular function $u_0(\text{nx})$ the Initial Boundary Value Problem (IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical solution $u=u(\text{nx},t)$ on $\mathrm{\Omega }\times I$: \begin{itemize} \item [$▹$] when $p>q$ and in 2- and 3-dimensional domains, we determine a $\mathit{\text{lower}}\mathit{\text{bound}}\mathit{\text{of}}$ $t^{\ast }$ for those $u$ becoming unbounded in $L^{m(p-1)}(\mathrm{\Omega })$ at such $t^{\ast }$; \item [$▹$] when $p<q$ and in $N$-dimensional settings, we establish a $\mathit{\text{global}}\mathit{\text{existence}}\mathit{\text{criterion}}$ for $u$. \end{itemize}